I�T. J, EL,EC1'.RONIUb, 1975, VOL. 38, :N"O. 4, 485-512

Measurement of loop gain in feedback systemst

R. D. MlDDLgBROOKt

In the design of a feedba.c:k system it is desira.bk t-o mak(' expcrimpntal rneasuremonts of the loop gain as a funct,ion of frequency to f'-nsure that t-he physical system operates as analyt.ica\ly prr-dicted or, if not, to supply information upon which a designcorrec­t,ion can be brn,3ed. In high loop-gain systems it is desirable that the loop-gain measurement- be umde without opening the loop. This paper discusses practical methods of measuring and interpreting t.he results for loop gain of the closed-loop system by a voltage injection or a current-injection technique ; extension to tho caso in which the measurement can be made even though the syskm is unstable ; and extension to the case in which neither t,he voltage nor current-injection technique alone is a<lcquatc, hut in ·which a combination of both permits the true loop gain to be derived. Thesf' tPchniquc8 have been found useful not onl,\· in linear feedback systems but Hlso 111 dcscrihing-functlon analysis of switchiug-mo<lf' converters and regulators.

I. Introduction

In approaching a system design prohlem, the usual procedure is to makea preliminary pa.per design and analysis, then to build a breadboard and make experimental tests of the performance. _If discrepancies are found bet,veen the predicted and observed properties of the system, the analysis model is corrected in a manner suggested by the nature of the observed discrepancies, and the modified predicted performance again compared with the experimental results. Several iterations of the analysis---measurement-�-correction sequence may be required before final adoption of the system design.

When the system being designed incorporates a negative feedback loop, one of the important performance parameters to be predicted analytically and experimentally verified is the loop gain. This paper is concerned with experimental methods of making :•mch measurement:-; with emphasis on practical problems of accuracy and proper interpretation of the results. These techniques have been found useful not only in linear feedback systems, but also in the describing-function analysis of many types of switching-mode converters and regulators.

The method of measuring loop gain 1' by injection of either a test voltage or a test current into the loop is first reviewed. The important feature of this method is that the loop remains closed, so that operating points are not disturbed. Use of a narrow-band voltmeter permits loop-gain measurements to be made in high-gain systems, and also in systems in which there is a large amount of noise a-R, for example, in a s,vitching-mode regulator. A technique for determination of phase as ,vel1 as magnitude of the loop gain, with use of

Received 23 September 1974. i' This work is an extension of material presented at the ESTEC Spacecraft

Pow-er Conditioning Electronics Seminar, ESRIN Centre, "Frascat.i, Italy, 20-22 May 1974.

t Profcs1mr of Electrical Engineering, California Institute of Technology, Pasadena, California.

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M wsarernen/ of loop gain in feedback systems 487

developed by the dependent voltage generator and a signal (either voltage or current) at the impedance Z. (Notahon: independent generators are repre­sented by circles, dependent generators by squares.) If the loop is opened and a test voltage v, applied at point A,., and the resulting voltage v

u of the

dependent generator measured, then, by definition, the loop gain is given by T=vvfv:r Since the loop is open, a voltage v

z=v

.r+v

11 appears across the

break, as shown in Fig. 1 (a).

A,

---Vz --+ ,,

- + T= v;-

I I + z

', ,, ---

• • +

feed bock

(□)

A;

,, T = -:--

''

z ---

feedback

lbl

Figure 1. Sieasurement of loop gain T by opening the loop, (a) hy voltage ratio, (b) by current ratio.

Another ' appropriate point ' is at the output of a dependent current generator, as indicated in Fig. I (b). In this case, 'opening the loop ' implies short-circuiting the output of the dependent current generator, and the loop gain, by definition, is T =i11/iv where i11 is the dependent generator current resulting from the test current i

x applied at point A

1. Since the loop is open,

a current iz= (r +i11 flows in the common connection as shown in Fig. I (b).

In many practical situations it is inconvenient to open the feedback loop in order to make such loop-gain measurements because, particularly in high­ga.in systems, it is difficult to maintain proper d.c. operating conditions, and the system may saturate on noise. However, the voltage conditions of }fig. I (a) can be maintained without opening the loop by the simple expedient of making vz , instead of v.r, the independent test signal: as shown in Fig. 2 (a), the original closed-loop circuit topology is retained, and alt.hough v

x and v

11

are now both dependent quantities, the loop gain is still T-vufv,. Similarly, in the dufll method shown in Fig. 2 (b ), substitution of i, as the independent test signal permits the original circuit topology to be retained and the loop

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il1 easurement of loop gain in feedback systems 489

The signals HJ'

, u11

and u2 are phasors, a.nd so use of an ordinar,Y voltmeter gives directly only the magnitude [ T [ of the loop gain. Often, it is known that the system being measured is minimum-phase, and so a measurement of loop-gain magnitude is sufficient. However, if phase mea::-mrements <UP

desired, a phase-rearling voltmeter such as the Princeton Applied Research Two-Phase/Vector Lock-In Amplifier :VIodcl 129 may be used. This instru­ment has the additional advantage that it is phase-locked to the test oscillator signal, ancl so functions as a narrow-band tracking voltmeter capable of reading very small signals that would otherwise be buried in noise. Indeed, the narrow-banrl property may be essential in order to make even the magnitude measurements, because in a system with high loop gain, u/J>u

.,.

and the allmvable magnitude of u11

is limited by system overloading; con­sequently, u.r is very small and may only be extracted from noise by use of a narrow-band voltmeter. Also, the narrow-band property is especially advantageous in making loop-gain measurements on switching regulators, since otherwise the switching noise ,vould be likely to s,vamp out at least the smaller of t.he loop-gain test Rignals.

load MODULATORf---�

JlJlJl

r· -----7 ,-----7 I i, A; ·,,

I I A, ,, I +-

I I + -

I I

11 1 I z, Z2I Z2I - � �

I ,, ,, ,, I

j j I I

+

L __ - __ J L _____ J

current voltaoe injection injectiOn

Figure :1. Appropriate poir11� for \'oltagl' and for current injection to mmumrc loop gain of a simple s,vitching regulator.

An example of the application of the signal-injection technique for measure­ment of the loop gain of a simple s,vitching regulator is Rhown in Fig. 3. As indicated, suitable inject,ion points A

r and A i:

for voltage and current injection reHpectively can be found. The location A,. is suitable for voltage injection because the impedance Z

1 looking forward around the loop is much greater

than the impedance Z2

looking backward, so that vu approximates the voltage of an ideal voltage generator, as required for the model of Fig. 2 (a) to be valid. Conversely, the location A i: is suitable for current injection because here the opposite condition Z,> Z

1 obtains, so that i" approximates the

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4!JO R. lJ. Middlebrook

current of an idBal current generahJf (namely, the collcetor of a t,ram,iRt.or), as required for the model of Fig. 2 (b) to he valid. Altho11gh it provi<leR directly only ma.guitn<lc information, a wave a.nalyRcr such as on0 of the Hmvlett-Packard :302A Rcrie.s iR a. ver;y convenient, im;tru­rnent for use in the signal-injection inethod of loop-gain measurement. (f-lpohn InG3, Hewlett-Packard Applicrilion Note, 1965): In the' BFO. mork the im;tn1rnent overates a.s an adjustabk-frecp1Pncy oseillator ,vith a.n auto­rna.tfr:a.lly tracking narrow-band voltmeter, and t,hus provides both the tc�t signal and thfl required narrow-band voltmeter. 1\-lorcovcr, phase information can also be obtained by a simple indirect method. At any frequency, one mea.sureR not, onl,v tho magnitudes lu.,. I and ln!I I to give ITl= j,,,,_1 (:!) I ll,I

but. a]so the rnagnit,udc ]u.zl of the t,hird phasor, so tha.t b,v trigonornd,ricsolution of the phai:mr triangle one obtains

or, equivalently, LT"�+ 2 cos--1 J(111,l'-(lu,I-- 111111)')- 4luxl ln11 I

(4 n)

(4 b)

Although the sign given by cqn. (4) is ambiguouf.i, proper choice is usually obvious from the qua.litat,jve nature of the magnitude response and knO\vn properties of U1e loop. The- above forms apply when LT is in t,he range 0° <±/_'fl < 180° ; for the range 180° < ± f _T < 360°, t,he appropriate formis L 71 = .± (3(i0° •- 0), ,vhere 0 is tlw first or second quadrant angle given by eqn. (4). In the current-injection method, rneasuremAnt of the third phaRor (: isstraightforward, but in the voltage-injection method measurement of ·1':: directly is inconvenient because of the rerp1irenrnnt to floa.t tlw voltmeter. lnstea.<l (in either method) the phasor sum ln'° I = iu.,+ uul can be read fromthe addition of the signals u.r and u !I passing through unit,v (or equal) gainamplifiers, as inclicated in Fig . 4 (a). Equation (4) gives accurate resulti,.; for the loop-gain pha,,c;;c angle in the important fr-equeney range in the rn•ighbourhood of loop-gain crossover, when ]Tl�l. Hov,'ever, it i8 ill-uonditioued ,vhcn either l'Pj}>J or l'l'l�l. sinue then either 11111 1;,,lu,I and luJ;,,:lu11 1. or 111_,l;,,lu,,I and ln,l"'IU.,i•This condition i1-1 imrncdia.td.v oLviom, in graphical terms from tlw pha.sor dia.hrr;im illustrated in Fig. 4 (a), drawn for [TI > l clearly, small errors m the rncasnrcrnent of the magnitude ]u 11 I or I Uz I can lead to large errors in

IT. The ill-conditioning defect in the plia:-;e measurement of Fig. ·4 (a) ca,n easily be overcome-]- by use of unequal gains for u.r a,nd v. 11 , as shown in Fig. -\-l,'scfnl suggestions relating to t,his techniqut.· were rn;ulf' by Dr. Yuan Yu of'l'R\V Hystcms, lnc .. and D .. J. Pa�kanl of the California .ln1-1titutc of Technology a.11d H nghes A ircrnft Co.

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